Librandi’s Sundaram-Type Triangle Complement

Abstract

Librandi’s triangle complement maps every outside value to a prime 4h+5.

Definition 1.1 (The triangle row formula).

$$\forall m\in \mathbb{N},n\in \mathbb{N},T\left(m,n\right)=\lfloor \frac{2\cdot m\cdot n+m+n-2}{2}\rfloor$$

Formalization. D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.T (✓ std3).

Citation. Vincenzo Librandi (2012). OEIS A140869, Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1. URL: https://oeis.org/A140869.

Commentary.

For natural m and n, T is the displayed natural-number quotient. The subtraction and division are literal truncated natural operations; on m >= n >= 1 the numerator is at least two, so the formula has its intended value.

Definition 1.2 (Membership in the triangle).

$$\forall h\in \mathbb{N},InTriangle\left(h\right)\iff (\exists m\in \mathbb{N},n\in \mathbb{N},1\le n\wedge \left(n\le m\wedge T\left(m,n\right)=h\right))$$

Formalization. D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.InTriangle (✓ std3).

Citation. Vincenzo Librandi (2012). OEIS A140869, Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1. URL: https://oeis.org/A140869.

Commentary.

A natural h belongs to the triangle exactly when it is T(m,n) for natural coordinates with 1 <= n <= m.

Theorem 1.3 (The complement-prime theorem).

$$\forall h\in \mathbb{N},\left(\neg InTriangle\left(h\right)\right)\Rightarrow Prime\left(4\cdot h+5\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.librandi_a140869 (✓ std3). ∎

Resolves. Problems/oeis-a140869-sundaram-triangle-complement-prime (proved) by D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.librandi_a140869.

Source. Repository-derived.

Commentary.

Assume that 4h+5 is composite. A nontrivial divisor supplied by the factorization theorem gives two odd factors. Writing them as 2a+1 and 2b+1, ordering the half-factors, and normalizing their product identity produces T(a,b)=h, a contradiction. The converse is false at h = 2, 8, and 12.

References

• Truth anchor: D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.InTriangle
• Truth anchor: D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.T
• Truth anchor: D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.librandi_a140869